4 Answers
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Most likely, you finished Halo 3 on a non-legendary ending. Bungie is famous for attaching extended endings for their highest difficulty level (at least with the Halo series). Complete the Legendary ending and you will see a short depicting what happens to the Master Chief.

After he says that, the front half of the Pillar of Autumn (with the Arbiter piloting) jumps to Earth, and the back half, along with Master Chief and Cortana, appears to jump to some other location in space. The people on Earth believe Master Chief to be dead, but he survives and goes into cryo-sleep. The final shot of the ending shows that the half of the ship with Master Chief is near/heading towards an unknown planet.

This video shows the ending, starting from the point after he says "We'll make it" and there is a bright flash

More recent information released about Halo 4 reveals that

At some point after these events, Master Chief is awoken from cryo-sleep and they land on the planet. There, he confronts previously unseen creatures and Covenant.

Well according to the Halo Novels that I've read, He is taken into a Dyson sphere, which is a really small sphere, but inside it using slip-space technology (halo equivalent of hyperspace) there can be a world of any size that is in a slower time, and can be like that infinitely with more and more spheres inside. (sort of like Inception, but instead of time being fastest in lower levels, it's slowest in realt-time and increasingly slower. but it is an inverted sphere. In Halo 4 he returns from it though, and is once again the character you play as in campaign, although they are training the spartan-4s. (spartans= ancient greeks, spartan-2s = master chief and the other 74 spartan-2s although only about 40-50 survived the physical augmentaations, spartan-3s had 2 batches, 1st one had 300, at age 12 they were sent to attack a base, they won, but only 2 survived, 2nd batch, forerunner machines kill all but ten, 5 of who are put in really deep dyson spheres, and mortally wounded. spartan-4s don't know much about, but the multiplayer in halo 4 is supposedly their training to give more sense(didn't make sense why spartans and elites were on teams together and switched teams every match, and spartans fought spartans, now it does because it all trainee spartans versus each other in whatever their "trainers" decide (aka custom games host, or ranom team selection in matchmaking))

I was reading a few posts on bungie.net. There were a lot of theories- some perhaps way off- but after reading the ones that explained part of it, I'd say that the flash of light was the slipspace rupture exploding like they tend to do. Realizing he was practically screwed, MC made way for the cryo bay in hopes that cyogenic suspension prevents the effects of Halo. Over the years, MC stayed a Cheif-cicle, keeping him alive. Installation 00- along with the Halo ring- probably drifted away, and similarly, Requiem drifted there- possibley for the 3rd or 4th time, considering its supposed to be a planet that might orbit a star or something.